When \(\nabla _{x_i}U(x_i,x_j) \equiv 0\)  for all \(i,j\le N\)  with \(i\ne j\) , the
dynamics is based solely on self-propulsion, and the resulting configuration
of agents is called a flock [1]}. That is, all of the agents
when started in parallel, will translate with the same asymptotic velocity and
direction. In the attraction-repulsion model, a flock undergoes translating
motion at a constant speed \(\sqrt{\alpha /\beta }\) , and can withstand
sufficiently small perturbations [2]}. However, due to
translation symmetry, there will always be one eigenvalue in the linearization
about the flocking state equal to zero, which means the stability is neutral
at best. Hence small perturbations may destabilize the flocking dynamics [3]}.
